The decomposition of the hypermetric cone into L-domains
aa r X i v : . [ m a t h . C O ] A ug THE DECOMPOSITION OF THE HYPERMETRIC CONE INTO L -DOMAINS MATHIEU DUTOUR SIKIRI ´C AND VIATCHESLAV GRISHUKHIN
Abstract.
The hypermetric cone HYP n +1 is the parameter space of basicDelaunay polytopes of n -dimensional lattice. If one fixes one Delaunay poly-tope of the lattice then there are only a finite number of possibilities for thefull Delaunay tessellations. So, the cone HYP n +1 is the union of a finite set of L -domains, i.e. of parameter space of full Delaunay tessellations.In this paper, we study this partition of the hypermetric cone into L -domains. In particular, we prove that the cone HYP n +1 of hypermetrics on n + 1 points contains exactly n ! principal L -domains. We give a detaileddescription of the decomposition of HYP n +1 for n = 2 , , n = 5 (see Table 2). Remarkable properties of the root system D are key for the decomposition of HYP . Introduction An n -dimensional lattice L is a subgroup of R n of the form L = v Z + · · · + v n Z with ( v , . . . , v n ) a basis of R n . Let S ( c, r ) be a sphere in R n with center c andradius r . Then, S ( c, r ) is said to be a Delaunay sphere in the lattice L if thefollowing two conditions hold:(i) k v − c k ≥ r for all v ∈ L ,(ii) the set S ( c, r ) ∩ L has affine rank n + 1.The n -dimensional polytope P , which is defined as the convex hull of the set S ( c, r ) ∩ L , is called a Delaunay polytope of rank n . The Delaunay polytopes ofrank n form a face-to-face tiling of R n . The Voronoi polytope P V ( L ) of a lattice L is the set of points, whose closest element in L is 0. Its vertices are centersof Delaunay polytopes of L . The polytope P V ( L ) forms a tiling of R n undertranslation by L , i.e. it is a parallelohedron (see Figure 1).The cones S n> , S n ≥ are respectively the cone of positive definite, positive semi-definite n × n matrices. The rational closure S nrat ≥ of S n> is defined as thepositive semidefinite matrices, whose kernel is defined by rational equalities (see[DSV06]). Given a basis B = ( v i ) ≤ i ≤ n of a lattice L , we associate the Gram Mathematics Subject Classification.
The Delaunay polytopes of L The Voronoi polytope of L and its translates Figure 1.
A lattice L ⊂ R and the induced partitionsmatrix a = B T B ∈ S n> . On the other hand if a ∈ S n> , then we can find an in-vertible real matrix B such that a = B T B , i.e. B is the basis of a lattice L = B Z n with Gram matrix a . So, we can replace the study of Delaunay polytopes of L for the standard scalar product by the study of Delaunay polytopes of Z n for thescalar product x T ax . If one takes another basis B ′ of L , then B ′ = BP for some P ∈ GL n ( Z ) and one has a ′ = P T aP , i.e. a and a ′ are arithmetically equivalent .In other words, the study of n -dimensional lattices up to isometric equivalence isthe same as the study of positive definite n × n symmetric matrices, up to arith-metic equivalence. In [DSV06] it is proved that if a ∈ S n ≥ , then one can define,possibly infinite, Delaunay polytopes of Z n for x T ax if and only if a ∈ S nrat ≥ .Given a polytope P of Z n the condition that it is a Delaunay polytope for thenorm x T ax translates to linear equalities and strict inequalities on the coefficientsof a . An L -domain is the convex cone of all matrices a ∈ S n> such that Z n has thesame Delaunay tessellation for x T ax (Details see, for example, in [Vo08, DL97,DSV06]). Voronoi proved that the cone S n> is partitioned into polyhedral L -domains. An L -domain of maximal dimension n ( n + 1) is called primitive . An L -domain is primitive if and only if the Delaunay tiling related to it consistsonly of simplices. Each non-primitive L -domain is an open face of the closureof a primitive one. In particular, an extreme ray of the closure of an L -domainis a non-primitive one-dimensional L -domain. The group GL n ( Z ) acts on the L -domains of S n> by D 7→ P T D P , and there is a finite number of orbits of L -domains, called L -types . The geometric viewpoint is most useful for thinking,and drawings about lattice and the Gram matrix viewpoint is the most suitableto machine computations.A metric on the set { , , . . . , n } is a function d such that d ( x, x ) = 0, d ( x, y ) = d ( y, x ) and d ( x, y ) ≤ d ( x, z ) + d ( z, y ). A metric d is an hypermetric if it satisfiesthe inequalities(1) H z ( d ) = X ≤ i 1) (triangle inequality)4 (1 , , − , , , − , , 0) and (1 , , , − , − 1) (pentagonal inequality)6 (1 , , − , , , , , , − , − , , , , , − , − 2) and (2 , , , − , − , − Table 1. The facets of HYP n for n ≤ n ) acts on HYP n ; it is proved in [DGP06] that there is noother symmetries if n = 4. It is proved in [DL97] that HYP n +1 is polyhedral,i.e. among the infinite set of inequalities of the form (1), a finite number sufficesto get all facets. This result can be proved in many different ways, see [DL97,Theorem 14.2.1]; the second proof uses that the image ξ (HYP n +1 ) is the union ofa finite number of L -domains. The purpose of this article is to investigate suchdecompositions of HYP n +1 .The set of orbits of facets of HYP n for n ≤ has 14orbits of facets (see [DL97, Ba99, RB]) and the list is not known for n ≥ 8. Aninequality of (1) is called k -gonal if P ni =0 | z i | = k . 3-gonal and 5-gonal inequalitiesare also called triangle and pentagonal inequalities, respectively.A Delaunay polytope P of a lattice L is called generating if the smallest, forthe inclusion relation, lattice containing V ( P ) is L . Moreover, if there exist afamily ( v , . . . , v n ) of vertices of P such that for any v ∈ L there exist α i ∈ Z with 1 = n X i =0 α i , v = n X i =0 α i v i then P is called basic and ( v , . . . , v n ) is an affine basis . Given such an affinebasis, we define the distance d ( i, j ) = k v i − v j k and we have H b ( d ) = X ≤ i Delaunay poly-topes. Note that if P = { , e , . . . , e n } , then the cone C n ( P ) is isomorphic to thecone HYP n +1 .The covariance map ξ : d → a transforms a hypermetric d on n + 1 points i ,0 ≤ i ≤ n , of a set X into an n × n positive semidefinite symmetric matrix a asfollows: a ij = ξ ( d ( i, j )) = 12 ( d (0 , i ) + d (0 , j ) − d ( i, j ))(see [DL97, Section 5.2]). The covariance ξ maps the hypermetric cone HYP n +1 into S nrat ≥ . Note that there are n + 1 distinct such maps depending on whichpoint of { , e , . . . , e n } is chosen as the zero point.2. Decomposition methods Recall that the Delaunay tiling related to a Gram matrix a from a primitive L -domain D consists of simplices. The set of Delaunay simplices of the tilingcontaining the common lattice point 0 is the star St . By translations, along Z n ,the star St determines fully the Delaunay tiling of Z n . The primitive L -domain D belongs to ξ (HYP n +1 ) if and only if its star St contains a main simplex.A wall W is an n ( n +1)2 − L -domain, which necessarily separatestwo primitive L -domains D , D ′ . Let one moves a point a from the primitive L -domain D to D ′ by passing through W . When a ∈ W , some pairs of simplices of St , which are mutually adjacent by a facet, glue into repartitioning polytopes. Itis well known (see, for example, [DSV06, Vo08]) that an n -dimensional polytopewith n + 2 vertices can be triangulated in exactly two ways. When the point a goes from W into the L -domain D ′ , each repartitioning polytope repartitionsinto its other set of simplices.Since each repartitioning polytope has n + 2 vertices, there is an affine depen-dence between its vertices. This affine dependence generates a linear equalitybetween the coefficients a ij of the Gram matrix a ∈ S n> . This equality is justthe equation determining the hyperplane supporting the wall W . If the point a lies inside the L -domain D , then this equality holds as an inequality.So, the convex hull of vertices of any pair of simplices of St adjacent by a facetis a putative repartitioning polytope giving an inequality separating the L -domain D from another L -domain. All adjacent pairs of simplices of St determine asystem of inequalities describing the polyhedral cone of the primitive L -domain D . Note that some of these inequalities define faces of D but not walls. If the HE DECOMPOSITION OF THE HYPERMETRIC CONE INTO L -DOMAINS 5 n L -domains L -types of HYP n +1 L -domains in HYP n +1 in HYP n +1 under Sym( n + 1)2 1 3 1 13 1 12 1 34 3 40 5 1725 222 210 8287 5 338 650 Table 2. Decomposition of HYP n +1 into L -domainsadjacent pair of simplices contains the main simplex then the corresponding walllies on a facet of the cone ξ (HYP n +1 ).Using the above system of inequalities, one can define all extreme rays of D ,and then all facets of D . The sum of Gram matrices lying on extreme rays of D is an interior point a ( D ) of D uniquely related to this L -domain (see [DSV06] formore details). Hence, L -domains D and D ′ belong to the same L -type if and onlyif a ( D ′ ) = P T a ( D ) P for some P ∈ GL n ( Z ), i.e. a ( D ′ ) and a ( D ) are arithmeticallyequivalent.The algorithm for enumerating primitive L -domains in ξ (HYP n +1 ) works asfollows. One takes a primitive Gram matrix a ∈ ξ (HYP n +1 ). There is a standardalgorithm which, for a given a ∈ S n> , constructs its simplicial Delaunay tiling.(For example, one can take a from a principal L -domain, described in followingsections).Using the star St of the Delaunay tiling, the algorithm, for each pair of ad-jacent simplices, determines the corresponding inequality. By the system of ob-tained inequalities, the algorithm finds all extreme rays of the domain D of a ,computes the interior central ray a ( D ) and finds all facets of D . The L -domain D is put in the list L of primitive L -domains in ξ (HYP n +1 ).Let F be a facet of D which does not lie on a facet of ξ (HYP n +1 ). For eachrepartitioning polytope related to the facet F of D , the algorithm finds anotherpartition into simplices. This gives the Delaunay tiling of the primitive L -domain D ′ , which is neighboring to D by the facet F . The algorithm finds all extreme raysof D ′ , the ray a ( D ′ ) and tests if it is arithmetically equivalent to a ( D ) for some D from L . If not, one puts D ′ in L . The algorithm stops when all neighboring L -domains are equivalent to ones in L . This algorithm is very similar to the onein [SY04] for the decomposition of the metric cone into T -domains and belongsto the class of graph traversal algorithms .We give some details of the partition of HYP n +1 into primitive L -domains inTable 2.We now expose another enumeration method of the orbits of primitive L -domains in HYP n +1 . Consider a primitive L -domain D . The group Stab( D ) = MATHIEU DUTOUR SIKIRI ´C AND VIATCHESLAV GRISHUKHIN { P ∈ GL n ( Z ) : P T a ( D ) P = a ( D ) } is a finite group, which permutes the trans-lation classes of simplices of the Delaunay decomposition of D . It splits the trans-lation classes of simplices into different orbits. Let S be a basic simplex in D ; ifone choose the coordinates such that vert S = { , e , . . . , e n } then one obtains an L -domain D S , whose image by ξ − is included in HYP n +1 . A permutation of thevertex set of S induces a permutation in HYP n +1 as well. So, two cones ξ − ( D S )and ξ − ( D S ′ ) are equivalent under Sym( n + 1) if and only if S and S ′ belongto the same orbit of translation classes of simplices under Stab( D ). Thereforefrom the list of L -types in dimension n , one obtains the orbits of L -domains inHYP n +1 .3. Dicings, rank extreme rays of an L -domain and of HYP n +1 We denote by b T c the scalar product of column vectors b and c . A vector v ∈ Z n is called primitive if the greatest common divisor of its coefficients is1; such a vector defines a family of parallel hyperplanes v T x = α for α ∈ Z .In the same way, a vector family V = ( v i ) ≤ i ≤ M of primitive vectors defines M families of parallel hyperplanes. A vector family V is called a lattice dicing iffor any n independent vectors v i , . . . , v i n ∈ V the vertices of the hyperplanearrangement v Ti j x = α i form the lattice Z n (see an example on Figure 1). Thisis equivalent to say that any n independent vectors v i have determinant ± 1, i.e.the vector family is unimodular . Given a dicing, the connected components ofthe complement of R n by the hyperplane arrangement form a partition of R n bypolytopes. It is proved in [ER94] that the polytopes of a lattice dicing defined by V = ( v i ) ≤ i ≤ M are Delaunay polytopes for the matrices belonging to the L -domaingenerated by the rank 1-forms ( v i v Ti ) ≤ i ≤ M , whose corresponding quadratic formis f v i ( x ) = ( v Ti x ) . The reverse is also proved there, i.e. any L -domain, whoseextreme rays have rank 1 has its Delaunay tessellation being a lattice dicing. Such L -domains are called dicing domains ; they are simplicial, i.e. their dimension isequal to their number of extreme rays.Our exposition of matroid theory is limited here to what is useful for thecomprehension of the paper and we refer to [DG99, Aig79, Tr92] for more details.Given a graph G of vertex-set { , . . . , n } we associate to every edge e = ( i, j )a vector v e , which is equal to 1 in position i , − j and 0 otherwise.The vector family V ( G ) = ( v e ) e ∈ E ( G ) is unimodular and is called the graphicunimodular system of the graph G . Given a unimodular n -dimensional system U of m vectors, for any basis B ⊆ U , we can write U = B ( I n , A ), where A isa totally unimodular matrix and ( I n , A ) is the concatenation of I n and A . Thematrix ( − A T , I m − n ) defines a unimodular system, which is called the dual of U and denoted Dual( U ). Given a graph G of vertex set V ( G ) and edge set E ( G ),we choose an orientation on every edge e and associate to it a coordinate x e .We define a vector space V to be the set of vector v ∈ R E ( G ) satisfying for all HE DECOMPOSITION OF THE HYPERMETRIC CONE INTO L -DOMAINS 7 x ∈ V ( G ) to the vertex cut equation X y ∈ N ( x ) v ( x,y ) ǫ ( x,y ) with N ( x ) the neighbors of x and ǫ ( x,y ) = 1 if the orientation of the edge ( x, y )goes from x to y and − v , . . . , v N a basis of the space anddenote by CoGr ( G ) the cographic unimodular system of the graph G defined tobe the vector system obtained by taking the transpose of the matrix ( v , . . . , v N ).The unimodular systems CoGr ( G ) and Dual( Gr ( G )) are isomorphic. In [DG99]a general method for describing unimodular vector families is given using graphic,cographic unimodular systems and a special unimodular system named E (or R as in [Se80]).Given a finite set E , a matroid M = M ( E ) is a family C ( M ) of subset of X called circuits such that: • for C , C ∈ C ( M ), it holds C C , C C if C = C ; • if e ∈ C ∩ C , then there is C ∈ C ( M ) such that C ⊆ C ∪ C − { e } .A set of vectors q e , e ∈ E , represents a matroid M ( E ) if, for any circuit C ∈C ( M ), the equality P e ∈ C q e = 0 holds. A matroid, is called regular if it admitsa representation as a unimodular system of vectors. If M ( E ) is a graphic or cographic matroid of a graph G with a set E of edges, then circuits of M are cycles or cuts of G , respectively.A graph G is plane if it is embedded in the 2-plane such that any two edges arenon-crossings. A plane graph defines a partition of the plane into faces delimitedby edges. The dual graph G ∗ is the graph defined by faces with an edge betweentwo faces if they share an edge. Then ( G ∗ ) ∗ = G , and there is a bijection between(intersecting) edges of G and G ∗ such that each cut of G corresponds to a cycle of G ∗ , and vice versa. In other words, the cographic matroid of G and the graphicmatroid of G ∗ are isomorphic.The only rank 1 extreme rays of the cone HYP n +1 are cut metrics . For n ≤ n +1 coincides with the cut cone CUT n +1 which is thecone hull of 2 n − N = { , . . . , n } ; if S ⊂ N , S = ∅ thenthe cut metric δ S on X = { } ∪ N is defined as follows: δ S ( i, j ) = 1 if |{ ij } ∩ S | = 1 , and δ S ( i, j ) = 0 , otherwise . The covariance map ξ transforms the cut metric δ S into the following correlation matrix p ( S ) of rank 1: p ij ( S ) = 1 if { ij } ⊂ S, and p ij ( S ) = 0 , otherwise, where 1 ≤ i, j ≤ n. The quadratic form corresponding to the correlation matrix p ( S ) is f S ( x ) = ξ ( δ S )( x ) = X p ij ( S ) x i x j = ( X i ∈ S x i ) . MATHIEU DUTOUR SIKIRI ´C AND VIATCHESLAV GRISHUKHIN So, f S = f q with f q ( x ) = ( q T x ) and q = P i ∈ S b i := b ( S ) the incidence vector ofthe set S . In summary: Lemma 1. A vector q ∈ Z n determines an extreme ray f q of ξ (HYP n +1 ) if andonly if q = b ( S ) for some S ⊆ N , S = ∅ . By Lemma 1, if U determines a dicing domain in ξ (HYP n ), then the set ofcoordinates of vectors from U in the basis B forms a unimodular matrix with(0 , , U determines a dicing L -domain D ( U ), we have the following proposition: Proposition 1. Let D ( U ) be a dicing domain determined by a unimodular set U . The following assertions are equivalent:(i) D ( U ) lies in ξ (HYP n ) ;(ii) U is represented by a (0 , -matrix. The principal L -domain There is a unique primitive L -type, whose L -domains are dicing domains ofmaximal dimension n ( n + 1) ([KoZo77, Di72]). Voronoi calls this L -type prin-cipal . Each principal L -domain is simplicial and all its n ( n + 1) extreme rayshave rank 1. The set Q of vectors q determining extreme rays f q of a principal L -domain forms a maximal unimodular system. This system is the classical uni-modular root system A n representing the graphic matroid of the complete graph K n +1 on n + 1 vertices.In our case, when a principal L -domain is contained in ξ (HYP n +1 ), its extremerays belong to the set { p ( S ) : S ⊆ N, S = ∅} of extreme rays of the cone ξ (HYP n +1 ). Hence, the vectors q have the form b ( S ) for S ⊆ N . We shall findall subsets of these vectors representing the graphic matroid of K n +1 . We orientedges of K n +1 into arcs and relate a vector b ( S ) to each arc of the directed graph K n +1 such that, for any directed circuit C in K n +1 , the following equality holds(2) X e ∈ C ε e b ( S e ) = 0 . Here b ( S e ) is the vector related to the arc e and ε e = 1 if the directions of e and C coincide, and ε e = − 1, otherwise.Given a chain of equally directed arcs labeled by one-element set, a subchainof this chain determines a (0 , { b ( S ) : S ⊆ N } .This relation of vectors b ( S ) and arcs of K n +1 provides a labeling of arcs of K n +1 by subsets S ⊆ N . We call this labeling feasible if the corresponding set HE DECOMPOSITION OF THE HYPERMETRIC CONE INTO L -DOMAINS 9 of vectors b ( S ) gives a representation of the graphical matroid of K n +1 , i.e. (2)holds for each circuit C of K n +1 .Consider a k -circuit C = { e i : 1 ≤ i ≤ k } , whose arcs have the same directions.Suppose that, for 1 ≤ i ≤ k , S i is a label of the arc e i , and that this labeling isfeasible. Then the equality P ki =1 b ( S i ) = 0 holds. Since the coordinates of thevectors b ( S ) take (0 , K n +1 with a feasible labeling has no circuit, whose arcshave the same directions. Any finite directed graph with no circuit has at leastone source vertex (and a sink vertex as well).Now consider a directed 3-circuit C = { e , e , e } of a feasible labeled K n +1 .Then two arcs of C , say the arcs e , e , have directions coinciding with the di-rection of C , and the third arc e has opposite direction. If S i is a label of e i , i = 1 , , 3, then we have the equality b ( S ) + b ( S ) = b ( S ). This equality ispossible only if S ∩ S = ∅ and S ∪ S = S . Since any two adjacent arcs of acomplete graph belong to a 3-circuit, we obtain the following result: Lemma 2. Let two arcs e i and e j be adjacent in a feasible labeled graph K n +1 andhave labels S i and S j . Then S i ∩ S j = ∅ if the directions of these arcs coincide inthe -path [ e i , e j ] . If these directions are opposite, then either S i ⊂ S j or S j ⊂ S i . Let v be a source vertex of K n +1 and E ( v ) be the set of n arcs incident to v .Since all arcs of E ( v ) go out from v , any two arcs e, e ′ ∈ E ( v ) have oppositedirections in their 2-path [ e, e ′ ]. Let S ( v ) = { S e : e ∈ E ( v ) } . By Lemma 2, thefamily S ( v ) is a nested family of n mutually embedded distinct subsets. Thisimplies that the sets S ∈ S ( v ) and the arcs e ∈ E ( v ) can be indexed as S i , e i ,1 ≤ i ≤ n , such that S i is the label of e i and | S i | = i .For 2 ≤ i ≤ n , let g i be the arc of the graph K n +1 , which forms a 3-circuit withthe arcs e i − , e i ∈ E ( v ). Lemma 2 implies that the arc g i has the one-elementset S i − S i − as label, and the direction of g i coincides with the direction of e i − in their 2-path [ e i − , g i ]. Now, it is clear that the n arcs g i for 1 ≤ i ≤ n , where g = e , form an n -path, whose arcs have the same directions and are labeled byone-element sets. Recall that a non self-intersecting n -path in a graph with n + 1vertices is called a Hamiltonian path . We obtain the following result. Lemma 3. A feasible labeled complete directed graph K n +1 has a Hamiltonianpath such that all its arcs have the same directions and each arc has a one-elementlabeling set. Let { } ∪ N be the set of vertices of K n +1 , where the vertex 0 is the source.Let 0 , i , i , . . . , i n be the vertices of the Hamiltonian path π in Lemma 3. Thepath π defines uniquely an orientation and a feasible labeling of K n +1 as follows.The arc with end-vertices i j , i k , where 0 ≤ j < k ≤ n , is labeled by the set S jk = { i r : j + 1 ≤ r ≤ k } ⊆ N . If one reverse the above order, then one gets thesame family of sets, and the labeled graph K n +1 gives the same representation ofthe unimodular system A n . We have Lemma 4. Any representation of the graphic matroid of the complete graph K n +1 by vectors b ( S ) , S ⊆ N , S = ∅ , is determined by a complete order of the set N .Two opposite orders determine the same representation. Since there are n ! complete orders on an n -set, as a corollary of Lemma 4, weobtain our main result. Proposition 2. The cone ξ (HYP n +1 ) contains n ! distinct principal L -domains. So, each principal domain is determined by an order (and its reverse) of theset N . For the sake of definition, we choose the lexicographically minimal order O from these two orders. Let S ( O ) be the family of sets S ⊆ N , S = ∅ , suchthat elements of each set S determine a continuous subchain of the n -chain,corresponding to the order O . A principal domain determined by an order O ofthe set N has n ( n + 1) extreme rays p ( S ) for S ∈ S ( O ).Each face F of a dicing L -domain in ξ (HYP n +1 ) is uniquely determined by itsextreme rays p ( S ), all of rank 1. Set S ( F ) = { S ⊆ N : p ( S ) is an extreme ray of F } . Proposition 3. If n ≥ , then any two principal domains in ξ (HYP n +1 ) are notcontiguous by a facet.Proof. If two principal domains D ( O ) , D ( O ′ ) ⊂ ξ (HYP n +1 ) share a facet F , thenthey have n ( n + 1) − p ( S ) for S ∈ S ( F ). This impliesthat the families S ( O ) and S ( O ′ ) should differ by one element only. But, for anytwo distinct orders O and O ′ , the families S ( O ) and S ( O ′ ) differ at least by twosets, if n ≥ 4. For example, suppose O and O ′ differ by a transposition of twoelements i and j . Then there is at least one subchain in O containing i and notcontaining j , which is not a subchain of O ′ . The same assertion is true for theorder O ′ . (cid:3) The decompositions of HYP and HYP Since, for n = 2 and n = 3 there exists only one primitive L -type, namely, theprincipal L -type, Proposition 2 describes completely the decompositions of thecones ξ (HYP n +1 ) for n = 2 , F of HYP n +1 is described by an inequality (1). If n ≤ F is a facet of HYP n +1 , then z i ∈ { , ± } . Hence, we can denote triangleand pentagonal facets as F ( ij ; k ) and F ( ijk ; lm ), respectively. Here z i = z j = 1, z k = − z l = 0, l ∈ X − { ijk } , for the triangle facet, and z i = z j = z k = 1, z l = z m = − z r = 0 if r ∈ X − { ijklm } , for the pentagonal facet.Note that ξ ( δ S ) = p ( S ). For S = { ij . . . k } , set S = ij . . . k and p ( S ) = p ( ij . . . k ). n = . The cone ξ (HYP ) is three-dimensional and simplicial. There is onlyone order O = (12) with S ( O ) = { , , } . Its three extreme rays p (1) , p (2) and p (12) span a principal domain, i.e. ξ (HYP ) coincides with a principal domain. HE DECOMPOSITION OF THE HYPERMETRIC CONE INTO L -DOMAINS 11 n = . The cone ξ (HYP ) is six-dimensional and has seven extreme rays p ( i ), i = 1 , , p ( ij ), ij = 12 , , p (123) and 12 facets ξ ( F ( ij ; k )) for i, j, k ∈{ } ∪ N . Note that p (1) + p (2) + p (3) + p (123) = p (12) + p (13) + p (23) = p. Hence, the four-dimensional cone C = R + p (1) + R + p (2) + R + p (3) + R + p (123)intersects by the ray R + p the three-dimensional cone C = R + p (12) + R + p (13) + R + p (23). The ray R + p is an interior ray of both.By Proposition 2, ξ (HYP ) contains three principal domains. These three six-dimensional L -domains are determined by the three orders (123), (132) and (213)of N = { } . Denote the domain determined by the order ( ijk ) as D j , where j is the middle element of the order ( ijk ). Since the three one-element subsets { i } , i ∈ N , and the set N give continuous chains in all the three orders, the fourrays p (1) , p (2) , p (3) and p (123) are common rays of all the three domains D , D and D . Hence, the cone C is the common four-dimensional face of thesethree principal domains. The domain D i is the cone hull of C and two rays p ( ij )and p ( ik ). The four triangle facets ξ ( F ( jk ; i )) , ξ ( F ( jk ; 0)) , ξ ( F (0 i ; j )) , ξ ( F (0 i ; k ))of ξ (HYP ) are also facets of D i . The other two facets of D i separating thedomain D i from D j and D k are the cone hulls of C with the rays p ( ij ) and p ( ik ),respectively. 6. L -domains in ξ (HYP )A parallelohedron is an n -dimensional polytope, whose image under a transla-tion group forms a tiling of R n . Given a face F of a parallelohedron P , the set offaces of P which are translates of F is called the zone of P . For a parallelohedron P the Minkowski sum P + z ( q ) may not be a parallelohedron. A parallelohedron P is called free along a vector q and the vector q is called free for a parallelohedron P if the sum P + z ( q ) is a parallelohedron (see [Gr06] for more details on thisnotion).If P q = P + z ( q ) is an n -dimensional parallelohedron, then P q has a non-zerowidth along the line l ( q ) spanned by q . This means that the intersection of P q with a line parallel to l ( q ) is distinct from a point. In this case, the lattice L q of the parallelohedron P q has a lamina H , i.e. a hyperplane H such that H istransversal to l ( q ), the intersection L q ∩ H is an ( n − L q and each Delaunay polytope of L q lies between two neighbouring layers of L q parallel to L q ∩ H (see [DeG02]). If a Voronoi polytope has a non-zero withalong a line l , then the lamina H is orthogonal to l .6.1. Root lattice D . The lattice D n is defined as D n = { x ∈ Z n | n X i =1 x i ≡ } If { e i : 1 ≤ i ≤ n } is an orthonormal basis of Z n , then the set of shortest vectorsof D n is ± e i ± e j for 1 ≤ i < j ≤ n . It is the set of all facet vectors of P V ( D n ) andform an irreducible root system, which we also denote by D n . There are threetranslation classes of Delaunay polytopes in D n : the cross polytope β n whosevertex set is formed by all e ± e i for 1 ≤ i ≤ n , the half cube H n whose vertexset is { x ∈ { , } n | P ni =1 x i ≡ } and a second half cube H ′ n whosevertex set is { x ∈ { , } × { , } n − | P ni =1 x i ≡ } . The two halfcubes are equivalent under the automorphism group Aut( D n ) of the root lattice D n . It is proved in [Gr04, DeG03] that the Voronoi polytope P V ( D n ) of the n -dimensional root lattice D n is free only along vectors which are parallel to edgesof P V ( D n ). Thus there are 2 n + 2 n free vectors.It turns out that when n = 4 all Delaunay polytopes are isometric to thecross-polytope β and equivalent under Aut( D ). Any 2-face of β is containedin three Delaunay polytopes β of the Delaunay tessellation. This proves thatthe L -domains of D are not dicing domains. Furthermore, D is a rigid lattice (see [BaGr01]), i.e. its Delaunay tessellation determines the Gram matrix up to ascalar multiple. This means that D determine a 1-dimensional L -type and thatany L -domain containing it as an extreme ray is not a dicing domain.The free vectors of P V ( D ) are parallel and can be identified to the diagonalsof the cross polytopes. They are • ± e i with 1 ≤ i ≤ β • ( ± , ± , ± , ± 1) with even plus signs for H ≡ β • ( ± , ± , ± , ± 1) with odd plus signs for H ′ ≡ β Up to a factor √ 2, those 24 vectors are an isometric copy of the root system D ,which we denote by D , . The union D ∪ D , is the irreducible root system F (see [Hu90]).The Voronoi polytope P V ( D ) of the root lattice D is the regular polytope 24-cell, whose automorphism group is the Coxeter group W ( F ) of Schl¨afli symbol { , , } (see [Cox48]). Its number of vertices, 2-faces, 3-faces is 24, 96, 24. Eachfacet is an octahedron with four pairs of opposite and mutually parallel triangular2-faces. The facet vectors of the Voronoi polytope P V ( D ) are the 24 roots of theroot system D . The polytope P V ( D ) has 12 edge zones of mutually paralleledges representing D , and 16 face zones of mutually parallel triangular faces.Each edge zone contains 8 parallel edges, and each face zone contains six parallelfaces.We choose a basis of D and denote by a ( D ) the Gram matrix of D in thisbasis. The 24 vertices of P V ( D ) are given by D , .6.2. The L -domains containing a ( D ) . For n = 4, there are three primitive L -types of four-dimensional lattices: the principal type, and L -types called byDelaunay in [De29] types II and III. The ten-dimensional L -domains of these L -types are constructed as follows (cf., [DeG03]). HE DECOMPOSITION OF THE HYPERMETRIC CONE INTO L -DOMAINS 13 Each k -dimensional face of a principal domain relates to the graphic matroid ofa subgraph G ⊆ K on k edges. Hence each facet (of dimension 9) of a principaldomain relates to the graphic matroid of K − , i.e. the complete graph K without one edge. The Gram matrix a ( D ) of the root lattice D is an extremeray of L -domains of type II and III. The cone hull of a facet of a principal domainand of a ray of type a ( D ) is an L -domain of type II. Hence, any principal domainis contiguous in S > by facets only with L -domains of type II.An L -domain of type II has the following three types of facets. One dicingfacet by which it is contiguous in S > to a principal domain relates to the graphicmatroid of the graph K − . Each of two other types of facets is the cone hullof the ray of type a ( D ) and a dicing 8-dimensional face related to the graphicmatroids of K − × or of K − . Here each of these graphs is the completegraph K without two non-adjacent or two adjacent edges, respectively.The complete bipartite graph K ij is formed by two blocks S , S of vertices with | S | = i , | S | = j and two vertices adjacent if and only if they belong to differentblocks. An L -domain of type III is the cone hull of a ( D ) and a 9-dimensionaldicing facet related to the cographic matroid CoGr ( K ) of the bipartite graph K . Each 8-element submatroid of CoGr ( K ) is graphic and relates to the graph K − × . Hence, each other facet of an L -domain of type III is the cone hull of a ( D ) and a dicing 8-dimensional face related to K − × . In S > , this facet isa common facet of L -domains of types II and III.So, an L -domain of type II is contiguous in S > to L -domains of all threetypes. It is useful to note ([RB05, DV05]) that if f belongs to the closure D ofan L -domain of type II or III then the Voronoi polytope P V ( f ) of Z n under thequadratic form f is an affine image of the Minkovski sum X q ∈ U λ q z ( q ) + λP V ( D ) , λ q ≥ , λ ≥ , where U is the unimodular set of vectors related to rank 1 extreme rays of D ,and P V ( D ) is the Voronoi polytope of the root lattice D , whose form a ( D ) liesalso on extreme ray of D .6.3. Unimodular systems in D , . Let D (4) ⊂ D , be a subset of 12 rootschosen by one from each pair of opposite roots. The vector system D (4) ispartitioned into three disjoint quadruples Q i , i = 1 , , 3, of mutually orthogonalroots, given in Subsection 6.1.For what follows, we have to consider triples ( r , r , r ) of roots chosen by onefrom each quadruple Q i , i.e. r i ∈ Q i , i = 1 , , 3. Let t = ( r , r , r ) be such atriple. Let { ijk } = { } , i.e. these three indices are distinct. We have r i = 2, r Ti r j ∈ {± } . Hence, the vectors(3) r ij = r i − ( r Ti r j ) r j for ij = 12 , , , are roots of D , . Since r Tij r i = − r Tij r j = 1, one of two opposite roots ± r ij belongsto Q k , say r ij ∈ Q k . Hence r ∈ Q , r ∈ Q , r ∈ Q . We have two cases: (i) r ij belongs, up to sign, to the triple t , i.e. r ij = r k , for all pairs ij ;(ii) r ij does not belong to t , i.e. r ij = r k .In case (i), the vectors r i , i = 1 , , 3, are linearly dependent, and the triple t spans a 2-dimensional plane. We say that the triple t is of rank 2. Note thatany two roots r, r ′ from distinct quadruples determine uniquely the third root r ′′ = r − ( r T r ′ ) r ′ and that there are 16 distinct triples of rank 2.In case (ii), the roots r i of the triple t are linearly independent. We say thatthe triple t has rank 3. In this case the roots r ij for ij = 12 , , 31, are distinctand do not coincide with the roots r i , i = 1 , , 3. Moreover, it is not difficult toverify that the triple ( r , r , r ) has rank 2.Triples of rank 2 and 3 are realized in √ P V ( D ) as follows. The roots of atriple t of rank 2 are parallel to edges of a 2-face of √ P V ( D ). We say that atriple t of rank 2 forms a face of √ P V ( D ). Since a triple of rank 2 forms a faceof √ P V ( D ), the 16 triples of rank 2 relate to the 16 zones of triangular faces ofit.Let t ′ = ( r ′ , r ′ , r ′ ) be a triple of rank 3. The 6 vectors ± r ′ i , i = 1 , , 3, haveend-vertices in vertices of √ P V ( D ). From each pair ± r ′ i of opposite roots, wechoose a vector r i such that r Ti r j = 1 for ij = 12 , , 31. Then end-vertices of theroots r i are vertices of a face F of √ P V ( D ), and the triple t = ( r ij = r i − r j : ij = 12 , , 31) has rank 2 and forms the face F .For what follows, we need subsets U of D (4) which are maximal by inclusionsuch that P V ( D ) + P q ∈ U λ q z ( q ) is a parallelohedron. Note that each quadruple Q i is a basis of R . Obviously, it is a unimodular set. It is a maximal byinclusion unimodular subset of D (4), since any other vector of D (4) has half-integer coordinates in this basis. However, it is proved in [DeG03], that P V ( D ) + P q ∈ U λ q z ( q ) is not a parallelohedron if U ⊂ D (4) is a quadruple, and it is aparallelohedron for any other maximal unimodular subsets U ⊆ D (4). Of course,a maximal unimodular set U = Q i does not contain each quadruple Q i as asubset.We show below that maximal unimodular subsets in D (4) represent either thegraphic matroid of the graph K − or the cographic matroid of the graph K (see Figure 2).The graph K − is planar. Hence, the graphic matroid of K − is isomorphicto the cographic matroid of the graph ( K − ) ∗ . Both graphs ( K − ) ∗ and K are cubic graphs on six vertices and nine edges. These graphs have a Hamiltonian6-circuit C on six vertices v i , 1 ≤ i ≤ 6. Call C with the six edges ( v i , v i +1 ) bya rim , and the other three edges by spokes . The spokes are the following edges: e = ( v , v ), e = ( v , v ), e = ( v , v ) in ( K − ) ∗ , and e i = ( v i , v i +3 ), i = 1 , , K (see Figure 2). Note that the edges e and e do not intersect in ( K − ) ∗ and intersect in K . The described form of K is the graph Q of [DG99].Besides the planarity, the graph ( K − ) ∗ differs from K by the number of cutsof cardinality three. All cuts of cardinality 3 of K are the six one-vertex cuts, HE DECOMPOSITION OF THE HYPERMETRIC CONE INTO L -DOMAINS 15 v e j v f k v e l v f j v e k v f l g j g k g i Cographic matroid of K v v v v v e l e k f j f l f k v e j Cographic matroid of ( K − ) ∗ Figure 2. Two cographic matroidsi.e. cuts containing three edges incident to a vertex. The graph ( K − ) ∗ , besidesthe six one-vertex cuts, has a separating cut containing three non-adjacent edgesone of which is a spoke. (These are the edges ( v , v ), ( v , v ) and the spoke( v , v ) of the above description of ( K − ) ∗ .) Proposition 4. A maximal by inclusion unimodular subset U of D (4) , U = Q i , i = 1 , , , is obtained by a deletion from D (4) of a triple t = ( r , r , r ) where r i ∈ Q i , i = 1 , , . Then(i) if t has rank , then U represents the cographic matroid of the graph K ;(ii) if t has rank , then U represents the graphic matroid of the graph K − ,which is isomorphic to the cographic matroid of the dual graph ( K − ) ∗ .Proof. Since Q i U , for i = 1 , , U does not contain a triple t . We show thata deletion from D (4) of any triple gives a unimodular set.Denote by V ( r ) the set of all triples of rank 2 containing a root r ∈ D (4). If r ∈ Q i and r ′ ∈ Q j , then V ( r ) ∩ V ( r ′ ) is the unique triple of rank 2 containing r and r ′ if i = j , and | V ( r ) ∩ V ( r ′ ) | = 0 if i = j and r = r ′ .Let r i ∈ Q i , i = 1 , , 3, be the roots of the deleted triple t . Then | V ( r i ) | = 4 , | V ( r i ) ∩ V ( r j ) | = 1 , and | V ( r ) ∩ V ( r ) ∩ V ( r ) | = 0 or 1 . Here, whether 0 or 1 stays in the last equality depends on the triple t has rank 3or 2, respectively. By the inclusion-exclusion principle, we have | ∪ i =1 V ( r i ) | = X i =1 | V ( r i ) | − X ≤ i 1, respectively.The same assertion is true for the arc p .We show that each root r ∈ R belongs to exactly two triples of V . Let r ∈ Q k , then | V ( r ) ∩ V ( r k ) | = 0 and | V ( r ) ∩ V ( r i ) | = | V ( r ) ∩ V ( r j ) | = 1, where { ijk } = { } . We have two cases:(4) V ( r ) ∩ V ( r i ) = V ( r ) ∩ V ( r j ) or V ( r ) ∩ V ( r i ) = V ( r ) ∩ V ( r j ) . Suppose that the inequality in (4) holds for r . Since | V ( r ) | = 4, only twotriples from V ( r ) belong to V .Now, if the equality in (4) holds, then it implies that r = ( r T r i ) r i + ( r T r j ) r j .If t has rank 2, then, up to sign, this gives r = r k , which contradicts to r ∈ R .Hence, if t has rank 2, the inequality of (4) holds for all r ∈ R . We can set V = V , since V contains 16 − 10 = 6 triples. It is easy to verify that G isisomorphic to K .If t has rank 3, then r = r ij , where r ij is defined in equation (3). The roots r ij , ij = 12 , , 31 when the equality holds in 4, form a triple v of rank 2. Obviously, v ∈ V . We set V = V − { v } . Since the root r ij belongs to two triples v and( r ij , r i , r j ), the remaining triples of V ( r ij ) belong to V . We saw that each root r ∈ R , r = r ij for ij = 12 , , 31, belongs to two triples of R . Hence, in thecase when t has rank 3, the graph G is well defined. It is easy to verify that G isisomorphic to ( K − ) ∗ and v corresponds to a separating cut of cardinality 3.Arcs of G are labeled naturally by roots from the set R . This labeling gives arepresentation of the cographic matroid of G by vectors of R .Note that triples of rank 2 are equivalent under action of the automorphismgroup of P V ( D ). Similarly, all triples of rank 3 are equivalent under the au-tomorphism group of P V ( D ) extended by changing signs of roots. Hence, anyexplicit representations of the cographic matroids of the graphs K and ( K − ) ∗ for fixed triples of rank 2 and 3 proves that above labeling give representationsfor all pairs of triples of rank 2 and 3. (cid:3) L -types in ξ (HYP ) . From Proposition 4 we deduce that each L -domaincorresponding to D is contained in 64 = 4 different L -types. 48 of them are oftype II and 16 are of type III.Recall that P V ( D ) is free along lines spanned by roots of the root system D , ,vectors of which are parallel to diagonals of the cross-polytopes β of the Delaunay HE DECOMPOSITION OF THE HYPERMETRIC CONE INTO L -DOMAINS 17 partition of the lattice D . Note that bases related to the forms a ( D ) ∈ ξ ( HY P )contain a diagonal of a β . Lemma 5. Let the basis related to a ( D ) contains a diagonal q ∈ D , of across-polytope β . Then the L -domain of the parallelohedron P q = P V ( D ) + z ( q ) related to this basis, i.e. the L -domain of the form a ( D ) + λf q , does not belongto ξ ( HY P ) .Proof. The parallelohedron P q has a non-zero width along the line l ( q ) parallelto q . The lattice L q of P q has a lamina H which is orthogonal to q . The lamina H separate the cross-polytope β with a diagonal q into two Delaunay polytopes,each being a pyramid with a base β orthogonal to q and lying in H . These twopyramids have the end-points of the diagonal q as apexes. Hence vertices of thebasic simplex of a ( D ) + λf q belong to two distinct Delaunay polytopes. Thisimplies that the L -domain of L q does not belong to ξ ( HY P ). (cid:3) Proposition 5. The cone ξ (HYP ) contains principal L -domains, L -domains of type II and L -domains of type III, total L -domains.Proof. By Proposition 2, ξ (HYP ) contains 4! = 12 principal L -domains.The closure of an L -domain of types II and III is the convex hull of a ( D ) anda dicing facet F ( U ) related to a unimodular subset U ⊂ D , of 9 vectors thatare free for P V ( D ).Each tile of the Delaunay tiling of the root lattice D is the regular four-dimensional cross polytope β . Any affine base of β contains exactly the twovertices w, w ′ of a diagonal and 3 vertices v , v , v chosen from the other diago-nals. We have d β ( w, w ′ ) = 4 and 2 for all other pairs. There are (cid:0) (cid:1) = 10 waysto choose a pair { w, w ′ } in a 5 elements set so there are exactly 10 rays a i ( D )representing D in HYP .Every L -domain a i ( D ) is contained in the closure of 64 primitive L -domains D ( U ), but not all of them are included in ξ (HYP ). Each L -domain D ( U ) hasthe form D ( U ) = conv ( a i ( D ) + F ( U )), where the subset U ⊂ D , is obtainedby a deletion of a triple from D , . 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